Nowadays, artificial intelligence (AI) is playing an increasingly important role in human society. Running AI algorithms represented by deep learning places great demands on computational power of hardware. However, with Moore's Law approaching physical limitations, the traditional Von Neumann computing architecture cannot meet the urgent demand for promoting hardware computational power. The brain-inspired neuromorphic computing (NC) employing an integrated processing-memory architecture is expected to provide an important hardware basis for developing novel AI technologies with low energy consumption and high computational power. Under this conception, artificial neurons and synapses, as the core components of NC systems, have become a research hotspot. This paper aims to provide a comprehensive review on the development of oxide neuron devices. Firstly, several mathematical models of neurons are described. Then, recent progress of Hodgkin-Huxley neurons, leaky integrate-and-fire neurons and oscillatory neurons based on oxide electronic devices is introduced in detail. The effects of device structures and working mechanisms on neuronal performance are systematically analyzed. Next, the hardware implementation of spiking neural networks and oscillatory neural networks based on oxide artificial neurons is demonstrated. Finally, the challenges of oxide neuron devices, arrays and networks, as well as prospect for their applications are pointed out.